SELECTED CHAPTERS FROM ALGEBRA I. R. Shafarevich Preface

68 I. R. ShafarevichTHEOREM 1. If the sets M 1 , ... , M r are nite, then the set M 1 M ris also nite, and n(M 1 M r )=n(M 1 ) n(M r ).We shall rst prove the theorem for the case of two sets, i.e. when r = 2 thiswill be the induction basis. If M 1 = fa 1 ...a n g, M 2 = fb 1 ...b m g, then all thepairs (a i b j ) can be written in the form of a rectangle(2)(a 1 b 1 ) ... (a n b 1 )(a 1 b 2 ) ... (a n b 2 )... ... ...(a 1 b m ) ... (a n b m )The j-th row above contains pairs whose last element isalways b j . In each rowthenumber of pairs is equal to the number of all a i 's, i.e. it is n. The number of therows is equal to the number of all b j 's, i.e. it is equal to m. Hence, the number ofpairs is nm. Notice that the rectangle (2) resembles, in a way, Fig. 2. (A dierentline of reasoning would be to say that the set M 1 is equivalent to the set f1 ...ngor to the set of monomials fx x 2 ...x n g and that M 2 is equivalent to the setfy y 2 ...y m g. Then, n(M 1 M 2 ) is, as we have seen, the number of terms inthe right-hand side of (1). Putting x =1,y =1,we conclude that this number ofterms is nm.)The proof of the general case of r sets M 1 , ... , M r will be carried out byinduction on r. In each sequence (a 1 ...a r )weintroduce two morebrackets, andwrite it in the form ((a 1 ...a r;1 )a r ). Clearly, this does not alter the numberof the sequences. But the sequence ((a 1 ...a r;1 )a r ) is the pair (x a r ), wherex =(a 1 ...a r;1 ) can be considered to be an element ofthesetM 1 M r;1 .Hence, the set M 1 M r is equivalent to the set P M r , where P = M 1 M r;1 . We have proved that n(P M r )=n(P )n(M r ), and by the inductionhypothesis we have n(P ) = n(M 1 ) n(M r;1 ). Therefore, n(M 1 M r ) =n(M 1 ) n(M r;1 )n(M r ) and the proof is complete.Using Theorem 1 we can once more form the expression for the number ofdivisors of a positive integer n. Suppose that n has the cannonical representationn = p 11 pr r :In Section 3 of Chapter I we saw that the divisors of n can be written in the formm = p 11 pr rwhere i can take anyintegral value between 0 and i (formula (11) of Chapter I).In other words, the set of divisors is equivalent to the set of sequences ( 1 ... r )where i takes the above mentioned values. But this is exactly the product M 1 M r of the sets M i where M i is the set f0 1 ... i g. Since n(M i )= i +1,according to Theorem 1 the number of divisors is ( 1 +1)( 2 +1)( r +1). Thisformula was derived in a dierent way in Section 3, Chapter I.